Skip to main content
SpringerLink
Log in

Automated non-monotonic reasoning in System P

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

This paper presents a novel approach to automated reasoning in System P. System P axiomatizes a set of core properties that describe reasoning with defeasible assertions (defaults) of the form: if α then normally (usually or typically) β. A logic with approximate conditional probabilities is used for modeling default rules. That representation enables reducing the satisfiability problem for default reasoning to the (non)linear programming problem. The complexity of the obtained instances requires the application of optimization approaches. The main heuristic that we use is the Bee Colony Optimization (BCO). As an alternative to BCO, we use Simplex method and Fourier-Motzkin Elimination method to solve linear programming problems. All approaches are tested on a set of default reasoning examples that can be found in literature. The general impression is that Fourier-Motzkin Elimination procedure is not suitable for practical use due to substantially high memory usage and time consuming execution, the Simplex method is able to provide useful results for some of the tested examples, while heuristic approach turns out to be the most appropriate in terms of both success rate and time needed for reaching conclusions. In addition, the BCO method was tested on a set of randomly generated examples of larger dimensions, illustrating its practical usability.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramé, A., Habet, D., Toumi, D.: Improving configuration checking for satisfiable random k-sat instances. Ann. Math. Artif. Intell. 79(1-3), 5–24 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adams, E.w.: The logic of conditionals: An application of probability to deductive logic. vol. 86. Springer (1975)

  3. Benferhat, S., Saffiotti, A., Smets, P.: Belief functions and default reasoning. Artif. Intell. 122(1), 1–69 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Booth, R., Paris, J.B.: A note on the rational closure of knowledge bases with both positive and negative knowledge. J. Log. Lang. Inf. 7(2), 165–190 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Camazine, S., Sneyd, J.: A model of collective nectar source by honey bees: self-organization through simple rules. J. Theor. Biol. 149, 547–571 (1991)

    Article  Google Scholar 

  6. Chen, Y., Wan, H., Zhang, Y., Zhou, Y.: Dl2asp: Implementing default logic via answer set programming. In: European Workshop on Logics in Artificial Intelligence, pp. 104–116. Springer (2010)

  7. Cholewinski, P., Marek, V.W., Truszczynski, M.: Default reasoning system deres. KR 96, 518–528 (1996)

    Google Scholar 

  8. Cholewiński, P., Marek, V.W., Truszczyński, M., Mikitiuk, A.: Computing with default logic. Artif. Intell. 112(1-2), 105–146 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Davidović, T., Ramljak, D., Šelmić, M., Teodorović, D.: Bee colony optimization for the p-center problem. Comput. Oper. Res. 38(10), 1367–1376 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Davidović, T., Teodorović, D., Šelmić, M.: Bee colony optimization Part I: The algorithm overview. Yugoslav J. Oper. Res. 25(1), 33–56 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Djeffal, M., Drias, H.: Multilevel bee swarm optimization for large satisfiability problem instances. Intell. Data Eng. Autom. Learn.–IDEAL 2013, 594–602 (2013)

    Google Scholar 

  12. Dubois, D., Prade, H.: Possibilistic logic, preferential models, non-monotonicity and related issues. In: Proceedings of the 12th International joint conference on Artificial Intelligence, IJCAI ’91, vol. 91, pp. 419–424 (1991)

  13. Eiter, T., Lukasiewicz, T.: Default reasoning from conditional knowledge bases: Complexity and tractable cases. Artif. Intell. 124(2), 169–241 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fagin, R., Halpern, J.Y., Megiddo, N.: A logic for reasoning about probabilities. Inf. Comput. 87(1), 78–128 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  15. Friedman, N., Halpern, J.Y.: Plausibility measures and default reasoning. J. ACM 48(4), 648–685 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Geffner, H.: Default Reasoning: Causal and Conditional Theories, vol. 4. MIT Press, Cambridge (1992)

    Google Scholar 

  17. Geffner, H., Pearl, J.: Conditional entailment: bridging two approaches to default reasoning. Artif. Intell. 53(2), 209–244 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Georgakopoulos, G., Kavvadias, D., Papadimitriou, C.H.: Probabilistic satisfiability. J. Complex. 4(1), 1–11 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  19. Goldszmidt, M., Pearl, J.: On the consistency of defeasible databases. Artif. Intell. 52(2), 121–149 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Goldszmidt, M., Pearl, J.: Rank-based systems: a simple approach to belief revision, belief update, and reasoning about evidence and actions. In: Proceedings of the 3rd International conference on principles of knowledge representation and reasoning, KR ’92, vol. 92, pp. 661–672 (1992)

  21. Hansen, P., Jaumard, B.: Probabilistic Satisfiability. In: Handbook of Defeasible Reasoning and Uncertainty Management Systems, pp. 321–367. Springer (2000)

  22. Ikodinović, N., Rašković, M., Marković, Z., Ognjanović, Z.: A first-order probabilistic logic with approximate conditional probabilities. Log. J. IGPL 22(4), 539–564 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jovanović, D., Mladenović, N., Ognjanović, Z.: Variable neighborhood search for the probabilistic satisfiability problem. Metaheuristics, 173–188 (2007)

  24. Kavvadias, D., Papadimitriou, C.H.: A linear programming approach to reasoning about probabilities. Ann. Math. Artif. Intell. 1(1-4), 189–205 (1990)

    Article  MATH  Google Scholar 

  25. Keisler, H.j.: Elementary calculus: an infinitesimal approach. Prindle Weber & Schmidt (1986)

  26. Kilani, Y.: Comparing the performance of the genetic and local search algorithms for solving the satisfiability problems. Appl. Soft Comput. 10(1), 198–207 (2010)

    Article  Google Scholar 

  27. Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artif. Intell. 44(1), 167–207 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  28. Krüger, T.J., Davidović, T.: Empirical analyis of the bee colony optmization method on 3-SAT problem. Proc. 43rd Symposium on Operations Research. SYM-OP-IS 2016, 297–301 (2016)

    Google Scholar 

  29. Lehmann, D.: Another perspective on default reasoning. Ann. Math. Artif. Intell. 15(1), 61–82 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. Lehmann, D., Magidor, M.: What does a conditional knowledge base entail?. Artif. Intell. 55(1), 1–60 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. Liao, T., Aydın, D., Stützle, T.: Artificial bee colonies for continuous optimization: Experimental analysis and improvements. Swarm Intell 7 (4), 327–356 (2013)

    Article  Google Scholar 

  32. Lü, Z., Hao, J.K.: Adaptive memory-based local search for MAX-SAT. Appl. Soft Comput. 12(8), 2063–2071 (2012)

    Article  Google Scholar 

  33. Lukasiewicz, T.: NMPROBLOG. http://www.kr.tuwien.ac.at/staff/lukasiew/nmproblog.html. [Online; accessed 20-August-2020]

  34. Lukasiewicz, T.: Probabilistic default reasoning with conditional constraints. Ann. Math. Artif. Intell. 34(1-3), 35–88 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lukasiewicz, T.: Nonmonotonic probabilistic logics under variable- strength inheritance with overriding: Algorithms and implementation in nmproblog (2005)

  36. Lukasiewicz, T.: Weak nonmonotonic probabilistic logics. Artif. Intell. 168(1), 119–161 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Lukasiewicz, T.: Nonmonotonic probabilistic logics under variable-strength inheritance with overriding: complexity, algorithms, and implementation. Int. J. Approx. Reason. 44(3), 301–321 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  38. Makinson, D.: General Theory of Cumulative Inference. In: International Workshop on Non-Monotonic Reasoning, pp. 1–18. Springer (1988)

  39. Marchiori, E., Rossi, C.: A flipping genetic algorithm for hard 3-SAT problems. In: Proceedings of the Genetic and Evolutionary Computation Conference, vol. 1, pp. 393–400 (1999)

  40. Marković, G., Teodorović, D., Aćimović-Raspopović, V.: Routing and wavelength assignment in all-optical networks based on the bee colony optimization. AI Commun. 20(4), 273–285 (2007)

    MathSciNet  MATH  Google Scholar 

  41. Munawar, A., Wahib, M., Munetomo, M., Akama, K.: Hybrid of genetic algorithm and local search to solve max-sat problem using nvidia cuda framework. Genet. Program Evolvable Mach. 10(4), 391–415 (2009)

    Article  Google Scholar 

  42. Nelder, J.A., Mead, R.: A simplex method for function minimization. Comput. J. 7(4), 308–313 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  43. Nicolas, P., Saubion, F., Stéphan, I.: Gadel: a Genetic Algorithm to Compute Default Logic Extensions. In: ECAI, pp. 484–490 (2000)

  44. Nicolas, P., Saubion, F., Stéphan, I.: Heuristics for a default logic reasoning system. Int. J. Artif. Intell. Tools 10(04), 503–523 (2001)

    Article  Google Scholar 

  45. Nicolas, P., Schaub, T.: The Xray System: an Implementation Platform for Local Query-Answering in Default Logics. In: Applications of Uncertainty Formalisms, pp. 354–378. Springer (1998)

  46. Nikolić, M., Teodorović, D.: Empirical study of the bee colony optimization (BCO) algorithm. Expert Syst. Appl. 40(11), 4609–4620 (2013)

    Article  Google Scholar 

  47. Nikolić, M., Teodorović, D.: Empirical study of the Bee Colony Optimization (BCO) algorithm. Expert Syst. Appl. 40(11), 4609–4620 (2013)

    Article  Google Scholar 

  48. Nilsson, N.J.: Probabilistic logic. Artif. Intell. 28, 71–87 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  49. Ognjanović, Z., Kratica, J., Milovanović, M.: A genetic algorithm for satisfiability problem in a probabilistic logic: A first report: Symbolic and Quantitative Approaches to Reasoning with Uncertainty, pp. 805–816 (2001)

  50. Ognjanović, Z., Midić, U., Kratica, J.: A genetic algorithm for probabilistic SAT problem. Artif. Intell. Soft Comput.-ICAISC 2004, 462–467 (2004)

    MATH  Google Scholar 

  51. Ognjanović, Z., Midić, U., MladenoviĆ, N.: A hybrid genetic and variable neighborhood descent for probabilistic SAT problem. In: Hybrid Metaheuristics, pp. 42–53. Springer (2005)

  52. Ognjanović, Z., Rašković, M.: Some first-order probability logics. Theor. Comput. Sci. 247(1), 191–212 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  53. Pearl, J.: Probabilistic semantics for nonmonotonic reasoning: a survey. In: Proceedings of the 1st International conference on principles of knowledge representation and reasoning, KR ’89, vol. 92, pp. 669–710 (1989)

  54. Pearl, J., System, Z.: A natural ordering of defaults with tractable applications to nonmonotonic reasoning. In: Proceedings of the 3rd Conference on Theoretical Aspects of Reasoning about Knowledge, pp. 121–135. Morgan Kaufmann Publishers Inc (1990)

  55. Rana, S., Whitley, D.: Genetic Algorithm Behavior in the MAXSAT Domain. In: Parallel Problem Solving from Nature—PPSN V, pp. 785–794. Springer (1998)

  56. Rašković, M., Marković, Z., Ognjanović, Z.: A logic with approximate conditional probabilities that can model default reasoning. Int. J. Approx. Reason. 49(1), 52–66 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  57. Reiter, R.: On reasoning by default. In: Proceedings of the 1978 workshop on Theoretical issues in natural language processing, pp. 210–218. Association for Computational Linguistics (1978)

  58. Reiter, R.: A logic for default reasoning. Artif. Intell. 13(1), 81–132 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  59. Robinson, A.: “Non-standard analysis” Studies in Logic and the Foundations of Mathematics (1966)

  60. Schaub, T.: A new methodology for query answering in default logics via structure-oriented theorem proving. J. Autom. Reason. 15(1), 95–165 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  61. Schaub, T., Linke, T., Brüning, S., Nicolas, P.: XRay: User’s Guide & Reference Manual. http://www.cs.uni-potsdam.de/wv/xray/. [Online; accessed 20-August-2020]

  62. Selman, B., Kautz, H.A., Cohen, B.: Noise Strategies for Improving Local Search. In: AAAI, vol. 94, pp. 337–343 (1994)

  63. Stojanović, T., Davidović, T., Ognjanović, Z.: Bee colony optimization for the satisfiability problem in probabilistic logic. Appl. Soft Comput. 31(0), 339–347 (2015)

    Article  Google Scholar 

  64. Stützle, T., Hoos, H., Roli, A.: A Review of the Literature on Local Search Algorithms for MAX-SAT. Rapport Technique AIDA-01-02, Intellectics Group. Darmstadt University of Technology, Germany (2001)

    Google Scholar 

  65. Teodorović, D.: Bee Colony Optimization (BCO). In: Lim, C.P., Jain, L.C., Dehuri, S. (eds.) Innovations in Swarm Intelligence, pp. 39–60. Springer, Berlin (2009)

  66. Teodorović, D., Dell’Orco, M.: Mitigating traffic congestion: solving the ride-matching problem by bee colony optimization. Transport. Plan. Techn. 31, 135–152 (2008)

    Article  Google Scholar 

  67. Teodorović, D., Šelmić, M., Davidović, T.: Bee colony optimization Part II: The application survey. Yugoslav J. Oper. Res. 25(2), 185–219 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  68. Villagra, M., Barán, B.: Ant colony optimization with adaptive fitness function for satisfiability testing. Log. Lang. Inf. Comput., 352–361 (2007)

  69. Šelmić, M., Teodorović, D., Vukadinović, K.: Locating inspection facilities in traffic networks: an artificial intelligence approach. Transport. Plan. Techn. 33, 481–493 (2010)

    Article  Google Scholar 

  70. Winston, W.L., Venkataramanan, M., Goldberg, J.B.: Introduction to Mathematical Programming, vol. 1. Thomson/Brooks/Cole Duxbury, Pacific Grove (2003)

    Google Scholar 

Download references

Acknowledgements

This work was supported by the Serbian Ministry of Education, Science and Technological Development (Agreements No. 451-03-9/2021-14/200122, 451-03-9/2021-14/200104, 451-03-9/2021-14/200029) and by the Science Fund of the Republic of Serbia, Grant AI4TrustBC: Advanced Artificial Intelligence Techniques for Analysis and Design of System Components Based on Trustworthy BlockChain Technology.

Author information

Authors and Affiliations

  1. Faculty of Science, University of Kragujevac, Kragujevac, Serbia

    Tatjana Stojanović

  2. Faculty of Mathematics, University of Belgrade, Belgrade, Serbia

    Nebojša Ikodinović

  3. Mathematical Institute, Serbian Academy of Science and Arts, Belgrade, Serbia

    Tatjana Davidović & Zoran Ognjanović

Authors
  1. Tatjana Stojanović
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nebojša Ikodinović
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tatjana Davidović
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zoran Ognjanović
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tatjana Stojanović.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Stojanović, T., Ikodinović, N., Davidović, T. et al. Automated non-monotonic reasoning in System P. Ann Math Artif Intell 89, 471–509 (2021). https://doi.org/10.1007/s10472-021-09738-2

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10472-021-09738-2

Keywords

Search

Navigation